Impedance increases in cascade network



March 6, 1956 c. v. HINTON IMPEDANCE INCREASES IN CASCADE NETWORK 2 Sheets-Sheet l Filed June 19, 1951 v u u v JNVENTOR: GUET/5 V H//VTO/V Arr'r March 6, 1956 c. v. HINTON IMPEDANCE INCRASES IN CASCADE NETWORK 2 Sheets-Sheet 2 Filed June 19, 1951 I VUV- JNVENTOR: UHT/S V H//VTO/V Arr'y nited States Patent llVIPEDANCE INCREASES IN CASCADE NETWORK Curtis V. Hinton, Indianapolis, Ind., assigner to the United States of America as represented by the Secretary of the Navy Application June 19, 1951, Serial No. 232,439

Claims. (Cl. 23S-61) (Granted under Title 35, U. S. Code (1952), sec. 266) This invention relates to an electrical network adapted for use in a computer. More specifically, it relates to a network of the cascade type.

In analogue computers, mathematical functions are generally represented by resistors which are wound in a manner to present a variation of resistance corresponding to a particular function. To perform certain types of mathematical operations, a plurality of such resistor stages are connected together in cascade with the output of one resistor stage taken across the movable contact thereof, feeding the input of the next resistor stage. The position of the movable contact on one resistor is usually independent of the position of the contact of a related stage. Any variation of the resistor setting of one particular stage should not affect the output of the preceding stage.

The numerous methods used in the prior art to effect this desirable end suffered from the disadvantages that they were not sufficiently effective to prevent some loading error on the previous stage, they required resistors and other components which were impractical or expensive to construct and resulted in appreciable attenuation.

One of the objects of the invention therefore is to provide a novel arrangement of resistors which eliminates the aforesaid disadvantages.

A further object of the invention is to provide a novel and improved arrangement of resistors in cascade which provides the variation of a resistor in one stage from affecting the output of the previous stage where the resistor may be of relatively simple and economical construction.

In a cascade network having a plurality of resistors designated to be varied and to represent a plurality of variable functions, it is desirable to employ compensating resistors so that the variation of a given resistor will truly indicate the variation of the function.

A still further object of the present invention is to pro vide improvements in a cascade electrical network, whereby the network is made practical and can handle a very wide range of functions upon which computations are to be made. With the present invention the construction and design of computer elements is amplified; there is less need for supplemental equipment; the output is increased by elimination of inherent attenuation; accuracy is increased by the simplication; and in fact, the method employed is theoretically accurate.

A broad feature of the invention is in providing a plurality of cascaded resistor network stages each cornprising a variable resistor wound to represent a desired function and a compensating variable resistor connected in series therewith across the input to the stage in a manner that the sum total of the resistance contributed by the function resistor of the stage at any setting thereof plus the parallel resistance combination of the compensating resistor associated therewith and the load of all the succeeding stages always equals the maximum value of the function resistor. The ratio of the maximum value of the function resistor of one stage to that of the preceding stage is made greater than one, so that the 2,737,343 Patented Mar. 6, 1956 tice value of the variable resistors of each stage is of reasonable value.

Other objects and many of the attendant advantages of this invention will be readily appreciated as the same becomes better understood by reference to the following detailed description when considered in connection with the accompanying drawings wherein:

Fig. 1 is a diagrammatic view of the cascade electrical network forming the invention of the present invention.

Fig. 2 shows function and compensating resistor cards used when a squared value is used in computation.

Fig. 3 shows the cards for use with a linear value.

Fig. 4 shows the cards for use with a square root.

Fig. 1 shows the electrical network of the present invention in which there are m variable function resistors 10 having maximum values R, RC1, RCiCz, RCiCzCa R(Cm-1! and a final or load resistor 11 having a fixed R(Cm)!, each of C1C2C3, and Cm being some constant that is greater than one. In the above expressions (Cm)!=(C1C2C3 Cm) and (Cm1)!=(C1CzC3 Cnt-1). Each of the Cs may be equal to any or all of the other Cs or to none of them, and all of the Cs may be equal. Each of these resistors is greater than the preceding resistor and is in fact a certain C times greater. The effective value of any function resistor is l-X) times its maximum value, X varying from l to 0, and is determined by the position of a shiftable tie or brush 12 extending between the function resistor 10 and its associated compensating resistor 13 and connected in series with the next function resistor R. Each compensating resistor is so calibrated that if the effective value of the function resistor is R( l-X the effective value of the compensating resistor at that tlme is RCX C-X which has a maximum of pensating resistors increase in the same ratio, thus having the values The final resistor 11 is in series with the tie 12 for the last function resistor 10. The final resistor and all the compensating resistors 13 are in parallel with one another. With the values of the function resistors, their associated compensating resistors and the final resistor as stated, the effect is one of constant loading for each stage function and compensating resistors regardless of the position of the tie 12, that is the value of X, and the resistance or impedance of each stage is a certain C times that of the preceding stage. Thus the function and compensating resistors of stage n and the load resistor 11 impost on stage n-l the following loading according to Ohms law:

This value can be seen to be constant, and independent of X. Similarly, the loading imposed on the stage n-Z can be shown to be R(Cm 2)!, and the loading on the rst stage-is RC1. t

Proceeding from Expression 1 above, it appears'that the voltage `drop across anystage is proportional to X and is'not affected by the particular values of X at the 'other stages. Thus the voltage drop across the nai resistoror load 11 is proportional to the product of all the Xs.

By studying the above formula, it can be seen that characteristics of the'network shown 'in Figure 1 and just described is that the total-resistance which one stage presents to theprece'dingstage is constant irrespective of the 'particular setting ofthev ganged resistors of the stage. uThe function*andcompensating resistors of eachistage vary in opposite directions, so that when one resistance -is zero,-the-other is'contributing its mato'n mum resistance and thev sum of the function resistance at any settingplus the net resistance of the compensating resistance is parallel with the net load following it, is always equal to the 'maximum value of the function resistor. By increasing the load presented by each succeeding stagethe value of the compensatingresistors become reasonable. This, of course, results in the maximum value Vof the function resistor of each stage and its compensating resistorprogressively increasing in value. Since the function resistor of each stage reaches its maximum value when the compensating resistor of the same stage is zero, 'it should be clear that the net resistance of the maximum value of the function resistance of the next stage is equal'to the maximum value of the function resistor ofthe stage including said latter compensating resistor,

As previously stated,'each Chas some value greater If each C is 'arbitrarily' made equal to` 2, then the compensating resistorihas a maximum value of 2. Since the maximum value of thefunction resistor of the second stage is C times that 'of the rst stage,the former equals 2. Its compensating resistor has a' maximum value of 4. Stated in another Way, the maximum value of m function resistors are 20, 21, 22 2m1, and the maximum values 'of n compensatingresistors are 21, 22, 23 2m. 'The final or'load resistor has a value of 2m. This makes working out of the resistors for all the ystages e'asier'for they are allsome power/of 2 times the firstl function resistor.

EachC cannot be less'than one, for then the"com pensating resistor would have a value of 'This cannot be handled with which would be negative. simple circuits.

If C=l,'then the maximum valueof the coinpensat ing resistor would be infinity 'when X=1,and some very large and unsatisfactory quantity when-X is somewhere near l. Thus thefunction resistors must increase in maximum value from 'stage to stage.

If C is very# large, then the function resistors increase tremendously from stage'to stage, and compensating resistors, being always largerthan their function resistors, since `is always larger than one, must also increase tremendously.

This would markedly decrease power output at the tinal resistor 11.

If C approaches one, the function resistors do not increase very fast, but each compensating resistor, being equal to C-l becomes very large.

Some intermediate value of C ismosdesirable Awhich is low enough to prevent the function resistors from becoming too high and high enough to prevent the compensating resistor from becoming too high. The particular' value of C for a given stage will depend ou the type of function represented by the function resistor in of the stage. Suppose that the function and ycornpensating resistors are each formed of wire-wound cards. llractical considerations may determine an upper limit to the ratio of maximum cardfheig'htto minimum-card height, for the maximum may be'limited VVby space, and the minimum by strength of card which'mustV be suicient to stand up under a winding.

Suppose that in a 'ce 'ain Steger/here the function resistor is R( l-X) and the ratio of the neat function resister to the instant one is C, 'Xzin so thatl the value of the function resistor is .3( ll"'l), and that of the compensating resistor' is function resistor will be expressed by NRt/rk and the height of the compensating resistor card bv RnC2V 1 (2) Now consideration is to be given toseveral types cf values of n: n l, it=l, 'and 0 n l. As previously stated, X may vary from 'G to l, and V=X and in all three cases 11 isY positive, V may vary from G to l.

ln the case n l, n-l is positive or :greater than 0. C was previously stated asbein'g gre 'thanone. 'Thos when V=G, the above expression "for the compensating resistor height equals Zero. When Vl, the expression mustbe greater than zero. Thel conditiony i/:Q must be excluded, because the compensating resistor most have some minimum height'to have'suhicient'stre h tov carry the winding. if a certain maximu'rratio of maximum height to minimum height of compe ating rcsistorcartl is allowable, which under certain conditions is il, and a certain maximum value of V, say l, is necessary, then the lower the value of n, the `lower 'the value of C, fora given minimum of V, as is evident'fromlxpression 2. Stated in `another way, for a givenl value of C, the lower the value of n, the lower the Vrninirn1`irn".fa'lue of V. Stated in still another way, for a given value of n, the lower the minimum Value of V, the higher the'value of C.

For the'condition 71:1, xpression 2becomes 'A minimum value of V=O is permissible, since the last expression does notgo to zero. If C is made equal to2, V may vary from() to l without the aforementioned height ratio exceeding the permitted value of 4. As the minimum value of V goes up from zero, thevalue of kC decreases from two toward one.

For the condition 0 n l, n-1 becomes negative, and WIr-1 goes up from zero toward infinity, as V goes from one* toward zero. Ifl the i aforementioned` height' ratio yef 4 isnot to beexceeded, thelower thevalue of'i'nythe v5 higher the minimum value of V for a given value of C. For a given minimum value of V, the lower the value of n, the higher the value of C.

For the condition n 0, n-l is even more negative than the condition n 1, and V"1 goes up even faster from zero toward infinity as V goes from one toward zero. If a given height ratio is not to be exceeded, the more negative the value of n, the higher the minimum value of V for a given value of C.

In Fig. 2 function and compensating resistors for the condition 11:2 are illustrated. Thus the vaine f the function resistor 10 is R(l-V2). Since the derivative of this expression is -2RV, the function resistor comprises a card 14 of uniform slope wound with a wire 15.

The value of the compensating resistor is RCV2 CV2 with a derivative of 2RC2V Thus the compensating resistor 13 comprises a card 16 with a curved slope wound with a wire 17. `A straight line is indicated in dash dot on card 16 to show the Value of 2RV, which is what the curved slope of card 16 will approach as a limit as C approaches infinity.

In Fig. 3 the condition n=1 is illustrated. Since the function resistor 10 has the value R(l-V), and the derivative is a constant, the function resistor has a card 1S of uniform height wound with wire 19. Since the compensating resistor 13 has the value RCV C V and the derivative is the compensating resistor has a card 20 with a curved slope wound with Wire 21.

in Fig. 4 the condition 11:0.5 is illustrated. Here the function resistor 10 has a value R( l-V) with a derivative of The function resistor has a card 22 with a curved edge, wound with a wire 23. The compensating resistor 13 has the value RCV with a derivative of RC2 2V.s(C V.s)2

With the value of C chosen between l and 3 the compensating resistor has a card 24 wound with a wire 25, card having a curved edge with the minimum height between its ends. With the value of C above 3 the minimum height will be at the end representing V=l.

Thus far the discussion has been only of the effect that V has on the compensating resistor card, but the function resistor card will also determine the range of values for V.

In Fig. 2, where the height of the function resistor' card 14 is 2R V, the ratio of maximum and minimum values of V cannot exceed 4 if the permissible height cannot exceed 4. Thus if V is to go as high as l, it cannot go below .25.

in Fig. 3 the height of the function resistor card 18 has a uniform height, and so V may vary from 0 to l so far as this card is concerned.

In Fig.'4, where the height of the function resistor card 22 is Ji 2V.s

the values of V may vary from .0625 to l, and the permissible height ratio will not exceed 4.

It is also important that the slope of the card be not too great, for otherwise the wire wound thereon may slide unless steps are provided. This may infiuence the selection of a constant C and the permissible limits of V depending on the value of n being employed.

If the constant C is the same from each function resistor to the next and the ratio of maximum and minimum card heights need not be taken into consideration, the optimum value of the constant is dependent on the number of sets m of function and compensating resistors, as follows:

with respect to C and equating the zero. Thus the optimum value of C varies from 2 with two stages and approaches 1 as a limit as the number of stages grows infinite.

Obviously many modifications and variations of the present invention are possible in the light of the above teachings. It is therefore to be understood that within the scope of the appended claims the invention may be practiced otherwise than as specifically described.

The invention described herein may be manufactured and used by and for the Government of the United States of America for governmental purposes without the payment of any royalties thereon ortherefor.

I claim:

1. A cascade network comprising a plurality of variable function resistors each having a maximum value C times that of the preceding one, C being any finite constant greater than one, and being capable of variation from one function resistor to the next, a plurality of associated compensating resistors, a shiftable tie between each variable resistor and its associated compensating resistor such that as the variable resistor varies from a minimum to the said maximum value by equalling (l-X) times the maximum value, X being proportional to the function to be measured by the resistor and varying from l to 0, the compensating resistor varies from a maximum to a minimum by equalling CX l times the maximum value of the function resistor, means connecting the tie of each set of function and compensating resistors in series with the next function resistor, a final resistor having a value C times the maximum value of the last variable function resistor, means connecting the final resistor in series with the tie associated with the last variable resistor, and means connecting the final resistor and the compensating resistances in parallel with one another.

2. The cascade network specified in claim l, the constant C being equal to 2.

3. A cascade computer network comprising a plurality of function resistors, a source of electrical supply, the resistors differing from each other by a constant factor, a compensating resistor for each function resistor, a shiftable connector between each function resistor and its compensating resistor, the rst function resistor being connected at one end to the source of supply, each connector except the last one having a conductor extending to the next function resistor of next higher value and connecting them in series, each pair of function and compensating resistors being connected in parallel with the other pairs to the source of supply, the resistance of the function resistor increasing and that of its cornpensating resistor diminishing as its connector is shifted from one end to the other, and a load resistor connected by the conductor from the last connector also lconnected in parallel with the pairs of resistors to the source of supply.

4. An assembly of resistors for use in computers coneprising a plurality of stages of resistor networks in cascade arrangement, each stage including a series arrangement of two variable resistors across the input thereto, one resistor variable to represent the variation of a predetermined mathematical function, the other resistor variable with said one resistor but in an opposite direction thereto, so that the sum total of the function resistor plus the net resistance of parallel combinatie-n of the said other resistance and the load presented by the stages following same is constant for any setting of the resistors,

means coupling the said other resistor across the input` of the succeeding stage, the net stage being greater than the preceding stage. Y

5. A plurality of resistor network stages in cascade arrangement wherein each stage includes a variable function resistor providing a resistance variation representing a mathematical function, a variable compensating resistor connected in series with said function resistor across the input to the stage, said compensating resistor wound so that the sum total of the resistance contributed by the function resistor of the stage at any setting thereof plus the net resistance of the compensating resistance in parallei with the load of all of the succeeding stages equals the maximum value of the function resistor of the stage, the values of the function resistors of each stage progressively increasing in value.

6. An assembly of resistors for use in computers comprising a plurality of stages of resistor networksin cascade arrangement; each stage including a series arrangement of two variable resistors across the input thereto; one of the resistors variable to represent the variation of a predetermined mathematical function, the other resistor variable with said one resistor but in an opposite direction thereto,

so that the sum total of the function resistors plus the net resistance of parallel combination of the said other resistance and the load presented by the stages following them is equal to the maximum value of the function resistor of that stage, means coupling the said other resistor across the input of the succeeding stage, the said other resistor having a maximum value such that the parallel combination of this resistor and the function resistor of the next stage is equal to the maximum value of the function resistor of the stage, the value of the function resistor of each stage being greater than the function resistance of the previous stage.

7. A plurality of resistor network stages in cascade arrangement wherein each stage includes a variable function resistor providing a resistance variation representing a' mathematical function, a variable compensating resistor connected in series with said function resistor across the input to the stage, means coupling the compensating resistor 0f each stage to the input of the succeeding stage, the function resistor of each stage being greater than the function resistor of the preceding stage, means varying said function and compensating resistors in opposite directions such that as the function resistor varies from a minimum to a maximum value by cqualling l-X times the maximum vaiue thereof, X being proportional to the function indicated by the resistor and variable from 1 to 0, the compensating resistor varies from a maximum to a minimum by equalling i Y times the maximum value of the function resistor of the stage where C is any number greater than 1 and is the ratio of the maximum value of the function resistor of the next stage to that of the function resistor of the stage, means coupling the compensating resistor of one stage to the input of the next succeeding stage.

8. The combination of claim 7 characterized further by a fixed load resistor coupled across the compensating resistor of the last stage and exceeding in value of the function resistor of the last stage, the net parallel resistance of the maximum value of the compensating resistance of the last stage and the said load resistor equalling the maximumvalue of the function resistor of the last stage.

9. The combination of claim 8 characterized further by C being equal to 2.

10. The combination of claim 7 characterized further by said coupling means providing an output Contact mov- Y able over the variable function resistor, and the variable References Cited in the file of this patent UNITED STATES PATENTS 2,114,330 Borden Apr. 19, 1.938 2,399,726 Doyle et al. May 7, 1946 2,410,651 Glass Nov. 5, 1946 OTHER REFERENCES Electronic Instruments, chapter 5, Radiation Lab. Series No. 2l, 1st edition, 1948, McGraw-Hill.

RCA Review, July 1938, RCA Institutes Technical Press, New York, pages 86-96.

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